ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 23 Jun 2024 19:10:46 GMT2024-06-23T19:10:46Z50421- On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum stateshttps://scholarbank.nus.edu.sg/handle/10635/116107Title: On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states
Authors: Matthews, W.; Winter, A.
Abstract: Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d x d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. This is a shortened version of a paper [1] recently submitted to Communications in Mathematical Physics; here the proofs have been omitted. ©2008 IEEE.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1161072008-01-01T00:00:00Z
- Weak decoupling duality and quantum identificationhttps://scholarbank.nus.edu.sg/handle/10635/115357Title: Weak decoupling duality and quantum identification
Authors: Hayden, P.; Winter, A.
Abstract: If a quantum system is subject to noise, it is possible to perform quantum error correction reversing the action of the noise if and only if no information about the system's quantum state leaks to the environment. In this paper, we develop an analogous duality in the case that the environment approximately forgets the identity of the quantum state, a weaker condition satisfied by Ε-randomizing maps and approximate unitary designs. Specifically, we show that the environment approximately forgets quantum states if and only if the original channel approximately preserves pairwise fidelities of pure inputs, an observation we call weak decoupling duality. Using this tool, we then go on to study the task of using the output of a channel to simulate restricted classes of measurements on a space of input states. The case of simulating measurements that test whether the input state is an arbitrary pure state is known as equality testing or quantum identification. An immediate consequence of weak decoupling duality is that the ability to perform quantum identification cannot be cloned. We, furthermore, establish that the optimal amortized rate at which quantum states can be identified through a noisy quantum channel is equal to the entanglement-assisted classical capacity of the channel, despite the fact that the task is quantum, not classical, and entanglement-assistance is not allowed. In particular, this rate is strictly positive for every nonconstant quantum channel, including classical channels. © 2012 IEEE.
Sun, 01 Jan 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1153572012-01-01T00:00:00Z
- Zero-error channel capacity and simulation assisted by non-local correlationshttps://scholarbank.nus.edu.sg/handle/10635/116180Title: Zero-error channel capacity and simulation assisted by non-local correlations
Authors: Cubitt, T.S.; Leung, D.; Matthews, W.; Winter, A.
Abstract: The theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly in the presence of various classes of nonsignalling correlations between sender and receiver i.e., shared randomness, shared entanglement and arbitrary nonsignalling correlations. When the channel being simulated is noiseless, this is zero-error coding assisted by correlations. When the resource channel is noiseless, it is the reverse problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, separations between the power of the different classes of assisting correlations are exhibited for finite block lengths. The most striking result here is that entanglement can assist in zero-error communication. In the large block length limit, shared randomness is shown to be just as powerful as arbitrary nonsignalling correlations for exact simulation, but not for asymptotic zero-error coding. For assistance by arbitrary nonsignalling correlations, linear programming formulas for the asymptotic capacity and simulation rates are derived, the former being equal (for channels with nonzero unassisted capacity) to the feedback-assisted zero-error capacity derived by Shannon. Finally, a kind of reversibility between nonsignalling-assisted zero-error capacity and exact simulation is observed, mirroring the usual reverse Shannon theorem. © 2011 IEEE.
Mon, 01 Aug 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1161802011-08-01T00:00:00Z
- "Hyperbits": The information quasiparticleshttps://scholarbank.nus.edu.sg/handle/10635/116899Title: "Hyperbits": The information quasiparticles
Authors: Pawłowski, M.; Winter, A.
Abstract: Information theory has its particles, bits and qubits, just as physics has electrons and photons. However, in physics we have a special category of objects with no clear counterparts in information theory: quasiparticles. They are introduced to simplify complex emergent phenomena making otherwise very difficult calculations possible and providing additional insight into the inner workings of the system. We show that we can adopt a similar approach in information theory. We introduce the hyperbits, information quasiparticles which we prove to be a resource equivalent to entanglement and classical communication, and we give examples of how they can be used to simplify calculations and get more insight into communication protocols. © 2012 American Physical Society.
Thu, 23 Feb 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1168992012-02-23T00:00:00Z
- All nonclassical correlations can be activated into distillable entanglementhttps://scholarbank.nus.edu.sg/handle/10635/116920Title: All nonclassical correlations can be activated into distillable entanglement
Authors: Piani, M.; Gharibian, S.; Adesso, G.; Calsamiglia, J.; Horodecki, P.; Winter, A.
Abstract: We devise a protocol in which general nonclassical multipartite correlations produce a physically relevant effect, leading to the creation of bipartite entanglement. In particular, we show that the relative entropy of quantumness, which measures all nonclassical correlations among subsystems of a quantum system, is equivalent to and can be operationally interpreted as the minimum distillable entanglement generated between the system and local ancillae in our protocol. We emphasize the key role of state mixedness in maximizing nonclassicality: Mixed entangled states can be arbitrarily more nonclassical than separable and pure entangled states. © 2011 American Physical Society.
Fri, 03 Jun 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1169202011-06-03T00:00:00Z
- The usefulness of uselessnesshttps://scholarbank.nus.edu.sg/handle/10635/115533Title: The usefulness of uselessness
Authors: Winter, A.
Thu, 26 Aug 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1155332010-08-26T00:00:00Z
- Entropic uncertainty relations-a surveyhttps://scholarbank.nus.edu.sg/handle/10635/43107Title: Entropic uncertainty relations-a survey
Authors: Wehner, S.; Winter, A.
Abstract: Uncertainty relations play a central role in quantum mechanics. Entropic uncertainty relations in particular have gained significant importance within quantum information, providing the foundation for the security of many quantum cryptographic protocols. Yet, little is known about entropic uncertainty relations with more than two measurement settings. In the present survey, we review known results and open questions. © IOP Publishing Ltd. and Deutsche Physikalische Gesellschaft.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/431072010-01-01T00:00:00Z
- Are random pure states useful for quantum computation?https://scholarbank.nus.edu.sg/handle/10635/115002Title: Are random pure states useful for quantum computation?
Authors: Bremner, M.J.; Mora, C.; Winter, A.
Abstract: We show the following: a randomly chosen pure state as a resource for measurement-based quantum computation is-with overwhelming probability-of no greater help to a polynomially bounded classical control computer, than a string of random bits. Thus, unlike the familiar "cluster states," the computing power of a classical control device is not increased from P to BQP (bounded-error, quantum polynomial time), but only to BPP (bounded-error, probabilistic polynomial time). The same holds if the task is to sample from a distribution rather than to perform a bounded-error computation. Furthermore, we show that our results can be extended to states with significantly less entanglement than random states. © 2009 The American Physical Society.
Mon, 11 May 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1150022009-05-11T00:00:00Z
- Pure-state transformations and catalysis under operations that completely preserve positivity of partial transposehttps://scholarbank.nus.edu.sg/handle/10635/115249Title: Pure-state transformations and catalysis under operations that completely preserve positivity of partial transpose
Authors: Matthews, W.; Winter, A.
Abstract: Motivated by the desire to better understand the class of quantum operations on bipartite systems that completely preserve positivity of partial transpose (PPT operations) and its relation to the class LOCC (local operations and classical communication), we present some results on deterministic bipartite pure-state transformations by PPT operations. Restricting our attention to the case in which we start with a rank K maximally entangled state, we give a necessary condition for transforming it into a given pure state, which we show is also sufficient when K is 2 and the final state has Schmidt rank 3. We show that it is sufficient for all K and all final states provided a conjecture about a certain family of semidefinite programs is true. We also demonstrate that the phenomenon of catalysis can occur under PPT operations and that, unlike LOCC catalysis, a maximally entangled state can be a catalyst. Finally, we give a necessary and sufficient condition for the possibility of transforming a rank K maximally entangled state to an arbitrary pure state by PPT operations assisted by some maximally entangled catalyst. © 2008 The American Physical Society.
Thu, 10 Jul 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1152492008-07-10T00:00:00Z
- Infinitely many constrained inequalities for the von neumann entropyhttps://scholarbank.nus.edu.sg/handle/10635/115150Title: Infinitely many constrained inequalities for the von neumann entropy
Authors: Cadney, J.; Linden, N.; Winter, A.
Abstract: We exhibit infinitely many new, constrained inequalities for the von Neumann entropy, and show that they are independent of each other and the known inequalities obeyed by the von Neumann entropy (basically strong subadditivity). The new inequalities were proved originally by Makarychev for the Shannon entropy, using properties of probability distributions. Our approach extends the proof of the inequalities to the quantum domain, and includes their independence for the quantum and also the classical cases. © 2012 IEEE.
Sun, 01 Jan 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1151502012-01-01T00:00:00Z
- Private capacity of quantum channels is not additivehttps://scholarbank.nus.edu.sg/handle/10635/115241Title: Private capacity of quantum channels is not additive
Authors: Li, K.; Winter, A.; Zou, X.; Guo, G.
Abstract: Recently there has been considerable activity on the subject of the additivity of various quantum channel capacities. Here, we construct a family of channels with a sharply bounded classical and, hence, private capacity. On the other hand, their quantum capacity when combined with a zero private (and zero quantum) capacity erasure channel becomes larger than the previous classical capacity. As a consequence, we can conclude for the first time that the classical private capacity is nonadditive. In fact, in our construction even the quantum capacity of the tensor product of two channels can be greater than the sum of their individual classical private capacities. We show that this violation occurs quite generically: every channel can be embedded into our construction, and a violation occurs whenever the given channel has a larger entanglement-assisted quantum capacity than (unassisted) classical capacity. © 2009 The American Physical Society.
Tue, 15 Sep 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1152412009-09-15T00:00:00Z
- Operational interpretations of quantum discordhttps://scholarbank.nus.edu.sg/handle/10635/112478Title: Operational interpretations of quantum discord
Authors: Cavalcanti, D.; Aolita, L.; Boixo, S.; Modi, K.; Piani, M.; Winter, A.
Abstract: Quantum discord quantifies nonclassical correlations beyond the standard classification of quantum states into entangled and unentangled. Although it has received considerable attention, it still lacks any precise interpretation in terms of some protocol in which quantum features are relevant. Here we give quantum discord its first information-theoretic operational meaning in terms of entanglement consumption in an extended quantum-state-merging protocol. We further relate the asymmetry of quantum discord with the performance imbalance in quantum state merging and dense coding. © 2011 American Physical Society.
Thu, 31 Mar 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124782011-03-31T00:00:00Z
- Quantum correlation without classical correlationshttps://scholarbank.nus.edu.sg/handle/10635/112493Title: Quantum correlation without classical correlations
Authors: Kaszlikowski, D.; Sen, A.; Sen, U.; Vedral, V.; Winter, A.
Abstract: We show that genuine multiparty quantum correlations can exist on its own, without a supporting background of genuine multiparty classical correlations, even in macroscopic systems. Such possibilities can have important implications in the physics of quantum information and phase transitions. © 2008 The American Physical Society.
Wed, 13 Aug 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124932008-08-13T00:00:00Z
- Information causality as a physical principlehttps://scholarbank.nus.edu.sg/handle/10635/112453Title: Information causality as a physical principle
Authors: Pawłowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Zukowski, M.
Abstract: Quantum physics has remarkable distinguishing characteristics. For example, it gives only probabilistic predictions (non-determinism) and does not allow copying of unknown states (no-cloning1). Quantum correlations may be stronger than any classical ones2, but information cannot be transmitted faster than light (no-signalling). However, these features do not uniquely define quantum physics. A broad class of theories exist that share such traits and allow even stronger (than quantum) correlations3. Here we introduce the principle of 'information causality' and show that it is respected by classical and quantum physics but violated by all no-signalling theories with stronger than (the strongest) quantum correlations. The principle relates to the amount of information that an observer (Bob) can gain about a data set belonging to another observer (Alice), the contents of which are completely unknown to him. Using all his local resources (which may be correlated with her resources) and allowing classical communication from her, the amount of information that Bob can recover is bounded by the information volume (m) of the communication. Namely, if Alice communicates m bits to Bob, the total information obtainable by Bob cannot be greater than m. For m = 0, information causality reduces to the standard nosignalling principle. However, no-signalling theories with maximally strong correlations would allow Bob access to all the data in any m-bit subset of the whole data set held by Alice. If only one bit is sent by Alice (m = 1), this is tantamount to Bob's being able to access the value of any single bit of Alice's data (but not all of them). Information causality may therefore help to distinguish physical theories from non-physical ones. We suggest that information causality - a generalization of the no-signalling condition - might be one of the foundational properties of nature. ©2009 Macmillan Publishers Limited. All rights reserved.
Thu, 22 Oct 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124532009-10-22T00:00:00Z
- Higher entropic uncertainty relations for anti-commuting observableshttps://scholarbank.nus.edu.sg/handle/10635/112449Title: Higher entropic uncertainty relations for anti-commuting observables
Authors: Wehner, S.; Winter, A.
Abstract: Uncertainty relations provide one of the most powerful formulations of the quantum mechanical principle of complementarity. Yet, very little is known about such uncertainty relations for more than two measurements. Here, we show that sufficient unbiasedness for a set of binary observables, in the sense of mutual anticommutation, is good enough to obtain maximally strong uncertainty relations in terms of the Shannon entropy. We also prove nearly optimal relations for the collision entropy. This is the first systematic and explicit approach to finding an arbitrary number of measurements for which we obtain maximally strong uncertainty relations. Our results have immediate applications to quantum cryptography. © 2008 American Institute of Physics.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124492008-01-01T00:00:00Z
- Quantum network communication-the butterfly and beyondhttps://scholarbank.nus.edu.sg/handle/10635/112499Title: Quantum network communication-the butterfly and beyond
Authors: Leung, D.; Oppenheim, J.; Winter, A.
Abstract: We study the problem of k-pair communication (or multiple unicast problem) of quantum information in networks of quantum channels. We consider the asymptotic rates of high fidelity quantum communication between specific sender-receiver pairs. Four scenarios of classical communication assistance (none, forward, backward, and two-way) are considered. (I) We obtain outer and inner bounds of the achievable rate regions in the most general directed networks. (II) For two particular networks (including the butterfly network), routing is proved optimal, and the free assisting classical communication can at best be used to modify the directions of quantum channels in the network. Consequently, the achievable rate regions are given by counting edge avoiding paths, and precise achievable rate regions in all four assisting scenarios can be obtained. (III) Optimality of routing can also be proved in classes of networks. The first class consists of directed unassisted networks in which (1) the receivers are information sinks, (2) the maximum distance from senders to receivers is small, and (3) a certain type of 4-cycles are absent, but without further constraints (such as on the number of communicating and intermediate parties). The second class consists of arbitrary backward-assisted networks with two sender-receiver pairs. (IV) Beyond the k-pair communication problem, observations are made on quantum multicasting and a static version of network communication related to the entanglement of assistance. © 2006 IEEE.
Thu, 01 Jul 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124992010-07-01T00:00:00Z
- Quantum rate-distortion coding with auxiliary resourceshttps://scholarbank.nus.edu.sg/handle/10635/112505Title: Quantum rate-distortion coding with auxiliary resources
Authors: Wilde, M.M.; Datta, N.; Hsieh, M.-H.; Winter, A.
Abstract: We extend quantum rate-distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression. The first setting we consider is that of quantum rate-distortion coding with the help of a classical side channel. Our result here is that the regularized entanglement of formation characterizes the quantum rate-distortion function, extending earlier work of Devetak and Berger. We also combine this bound with the entanglement-assisted bound from our prior work to obtain the best known bounds on the quantum rate-distortion function for an isotropic qubit source. The second setting we consider is that of quantum rate-distortion coding with quantum side information (QSI) available to the receiver. In order to prove results in this setting, we first state and prove a quantum reverse Shannon theorem with QSI (for tensor-power states), which extends the known tensor-power quantum reverse Shannon theorem. The achievability part of this theorem relies on the quantum state redistribution protocol, while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver's QSI. This quantum reverse Shannon theorem with QSI naturally leads to quantum rate-distortion theorems with QSI, with or without entanglement assistance. © 1963-2012 IEEE.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1125052013-01-01T00:00:00Z
- Quantum-to-classical rate distortion codinghttps://scholarbank.nus.edu.sg/handle/10635/112510Title: Quantum-to-classical rate distortion coding
Authors: Datta, N.; Hsieh, M.-H.; Wilde, M.M.; Winter, A.
Abstract: We establish a theory of quantum-to-classical rate distortion coding. In this setting, a sender Alice has many copies of a quantum information source. Her goal is to transmit a classical description of the source, obtained by performing a measurement on it, to a receiver Bob, up to some specified level of distortion. We derive a single-letter formula for the minimum rate of classical communication needed for this task. We also evaluate this rate in the case in which Bob has some quantum side information about the source. Our results imply that, in general, Alice's best strategy is a non-classical one, in which she performs a collective measurement on successive outputs of the source. © 2013 American Institute of Physics.
Tue, 02 Apr 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1125102013-04-02T00:00:00Z
- Entanglement of the Antisymmetric Statehttps://scholarbank.nus.edu.sg/handle/10635/112431Title: Entanglement of the Antisymmetric State
Authors: Christandl, M.; Schuch, N.; Winter, A.
Abstract: We analyse the entanglement of the antisymmetric state in dimension d × d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state. Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results. © 2012 Springer-Verlag.
Sun, 01 Apr 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124312012-04-01T00:00:00Z
- Counterexamples to additivity of minimum output p-rényi entropy for p close to 0https://scholarbank.nus.edu.sg/handle/10635/112404Title: Counterexamples to additivity of minimum output p-rényi entropy for p close to 0
Authors: Cubitt, T.; Harrow, A.W.; Leung, D.; Montanaro, A.; Winter, A.
Abstract: Complementing recent progress on the additivity conjecture of quantum information theory, showing that the minimum output p-Rényi entropies of channels are not generally additive for p > 1, we demonstrate here by a careful random selection argument that also at p = 0, and consequently for sufficiently small p, there exist counterexamples. An explicit construction of two channels from 4 to 3 dimensions is given, which have non-multiplicative minimum output rank; for this pair of channels, numerics strongly suggest that the p-Rényi entropy is non-additive for all p ≃ 0.11. We conjecture however that violations of additivity exist for all p < 1. © 2008 Springer-Verlag.
Sat, 01 Nov 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124042008-11-01T00:00:00Z
- Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1https://scholarbank.nus.edu.sg/handle/10635/112405Title: Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1
Authors: Hayden, P.; Winter, A.
Abstract: For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p > 1, the minimum output Rényi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Rényi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p = 1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2. © 2008 Springer-Verlag.
Sat, 01 Nov 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124052008-11-01T00:00:00Z
- Entanglement and separability of quantum harmonic oscillator systems at finite temperaturehttps://scholarbank.nus.edu.sg/handle/10635/112427Title: Entanglement and separability of quantum harmonic oscillator systems at finite temperature
Authors: Anders, J.; Winter, A.
Abstract: the present paper we study the entanglement properties of thermal (a.k.a. Gibbs) states of quantum harmonic oscillator systems as functions of the Hamiltonian and the temperature. We prove the physical intuition that at sufficiently high temperatures the thermal state becomes fully separable and we deduce bounds on the critical temperature at which this happens. We show that the bound becomes tight for a wide class of Hamiltonians with sufficient translation symmetry. We find, that at the crossover the thermal energy is of the order of the energy of the strongest normal mode of the system and quantify the degree of entanglement below the critical temperature. Finally, we discuss the example of a ring topology in detail and compare our results with previous work in an entanglement-phase diagram. © Rinton Press.
Sat, 01 Mar 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1124272008-03-01T00:00:00Z
- Identification via quantum channelshttps://scholarbank.nus.edu.sg/handle/10635/112550Title: Identification via quantum channels
Authors: Winter, A.
Abstract: We review the development of the quantum version of Ahlswede and Dueck's theory of identification via channels. As is often the case in quantum probability, there is not just one but several quantizations: we know at least two different concepts of identification of classical information via quantum channels, and three different identification capacities for quantum information. In the present summary overview we concentrate on conceptual points and open problems, referring the reader to the small set of original articles for details. © Springer-Verlag Berlin Heidelberg 2013.
Tue, 01 Jan 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1125502013-01-01T00:00:00Z
- A resource framework for quantum Shannon theoryhttps://scholarbank.nus.edu.sg/handle/10635/112378Title: A resource framework for quantum Shannon theory
Authors: Devetak, I.; Harrow, A.W.; Winter, A.J.
Abstract: Quantum Shannon theory is loosely defined as a collection of coding theorems, such as classical and quantum source compression, noisy channel coding theorems, entanglement distillation, etc., which characterize asymptotic properties of quantum and classical channels and states. In this paper, we advocate a unified approach to an important class of problems in quantum Shannon theory, consisting of those that are bipartite, unidirectional, and memoryless. We formalize two principles that have long been tacitly understood. First, we describe how the Church of the larger Hilbert space allows us to move flexibly between states, channels, ensembles, and their purifications. Second, we introduce finite and asymptotic (quantum) information processing resources as the basic objects of quantum Shannon theory and recast the protocols used in direct coding theorems as inequalities between resources. We develop the rules of a resource calculus which allows us to manipulate and combine resource inequalities. This framework simplifies many coding theorem proofs and provides structural insights into the logical dependencies among coding theorems. We review the above-mentioned basic coding results and show how a subset of them can be unified into a family of related resource inequalities. Finally, we use this family to find optimal tradeoff curves for all protocols involving one noisy quantum resource and two noiseless ones. © 2008 IEEE.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1123782008-01-01T00:00:00Z
- A non-distiLIability criterion for secret correlationshttps://scholarbank.nus.edu.sg/handle/10635/112377Title: A non-distiLIability criterion for secret correlations
Authors: Masanes, L.; Winter, A.
Abstract: Within entanglement theory there are criteria which certify that some quantum states cannot be distilled into pure entanglement. An example is the positive partial transposition criterion. Here we present, for the first time, the analogous thing for secret correlations. We introduce a computable criterion which certifies that a probability distribution between two honest parties and an eavesdropper cannot be (asymptotically) distilled into a secret key. The existence of non-distillable correlations with positive secrecy cost, also known as bound information, is an open question. This criterion may be the key for finding bound information. However, if it turns out that this criterion does not detect bound information, then, a very interesting consequence follows: any distribution with positive secrecy cost can increase the secrecy content of another distribution. In other words, all correlations with positive secrecy cost constitute a useful resource. © Rinton Press.
Fri, 01 Jan 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1123772010-01-01T00:00:00Z
- Highly entangled states with almost no secrecyhttps://scholarbank.nus.edu.sg/handle/10635/117038Title: Highly entangled states with almost no secrecy
Authors: Christandl, M.; Schuch, N.; Winter, A.
Abstract: In this Letter we illuminate the relation between entanglement and secrecy by providing the first example of a quantum state that is highly entangled, but from which, nevertheless, almost no secrecy can be extracted. More precisely, we provide two bounds on the bipartite entanglement of the totally antisymmetric state in dimension d×d. First, we show that the amount of secrecy that can be extracted from the state is low; to be precise it is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that we need a large amount of singlets to create the state: entanglement cost is larger than a constant, independent of d. In order to obtain our results we use representation theory, linear programming, and the entanglement measure known as squashed entanglement. Our findings also clarify the relation between the squashed entanglement and the relative entropy of entanglement. © 2010 The American Physical Society.
Thu, 17 Jun 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1170382010-06-17T00:00:00Z
- Random quantum codes from gaussian ensembles and an uncertainty relationhttps://scholarbank.nus.edu.sg/handle/10635/116560Title: Random quantum codes from gaussian ensembles and an uncertainty relation
Authors: Hayden, P.; Shor, P.W.; Winter, A.
Abstract: Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the senders input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement which, concluding, ensures the existence of a decoding by the receiver. © 2008 World Scientific Publishing Company.
Sat, 01 Mar 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1165602008-03-01T00:00:00Z
- The structure of Rényi entropic inequalitieshttps://scholarbank.nus.edu.sg/handle/10635/116647Title: The structure of Rényi entropic inequalities
Authors: Linden, N.; Mosonyi, M.; Winter, A.
Abstract: We investigate the universal inequalities relating the α-Rényi entropies of the marginals of a multipartite quantum state. This is in analogy to the same question for the Shannon and von Neumann entropies (α =1), which are known to satisfy several non-trivial inequalities such as strong subadditivity. Somewhat surprisingly, we find for 0 < α < 1 that the only inequality is non-negativity: in other words, any collection of non-negative numbers assigned to the non-empty subsets of n parties can be arbitrarily well approximated by the α-entropies of the 2n α 1 marginals of a quantum state. For α >1, we show analogously that there are no non-trivial homogeneous (in particular, no linear) inequalities. On the other hand, it is known that there are further, nonlinear and indeed non-homogeneous, inequalities delimiting the α-entropies of a general quantum state. Finally, we also treat the case of Rényi entropies restricted to classical states (i.e. probability distributions), which, in addition to non-negativity, are also subject to monotonicity. For α ≠ 0, 1, we show that this is the only other homogeneous relation. © 2013 The Author(s).
Tue, 08 Oct 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1166472013-10-08T00:00:00Z
- How many copies are needed for state discrimination?https://scholarbank.nus.edu.sg/handle/10635/112566Title: How many copies are needed for state discrimination?
Authors: Harrow, A.W.; Winter, A.
Abstract: A theorem has been proved to understand the fact that how many copies of the unknown quantum state does one need to be able to distinguish the coset state with high reliability. It is proved that for every distribution there exists a POVM attaining success probability. By using the game-theoretic trick, it has been proved that the existence of a measurement with worst-case.
Sun, 01 Jan 2012 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1125662012-01-01T00:00:00Z
- Quantum locking of classical correlations and quantum discord of classical-quantum stateshttps://scholarbank.nus.edu.sg/handle/10635/116552Title: Quantum locking of classical correlations and quantum discord of classical-quantum states
Authors: Boixo, S.; Aolita, L.; Cavalcanti, D.; Modi, K.; Winter, A.
Abstract: A locking protocol between two parties is as follows: Alice gives an encrypted classical message to Bob which she does not want Bob to be able to read until she gives him the key. If Alice is using classical resources, and she wants to approach unconditional security, then the key and the message must have comparable sizes. But if Alice prepares a quantum state, the size of the key can be comparatively negligible. This effect is called quantum locking. Entanglement does not play a role in this quantum advantage. We show that, in this scenario, the quantum discord quantifies the advantage of the quantum protocol over the corresponding classical one for any classical-quantum state. © 2011 World Scientific Publishing Company.
Sat, 01 Oct 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1165522011-10-01T00:00:00Z
- On the dimension of subspaces with bounded Schmidt rankhttps://scholarbank.nus.edu.sg/handle/10635/116493Title: On the dimension of subspaces with bounded Schmidt rank
Authors: Cubitt, T.; Montanaro, A.; Winter, A.
Abstract: We consider the question of how large a subspace of a given bipartite quantum system can be when the subspace contains only highly entangled states. This is motivated in part by results of Hayden [e-print arXiv:quant-ph0407049; Commun. Math. Phys., 265, 95 (2006)], which show that in large d×d -dimensional systems there exist random subspaces of dimension almost d2, all of whose states have entropy of entanglement at least log d-O (1). It is also a generalization of results on the dimension of completely entangled subspaces, which have connections with the construction of unextendible product bases. Here we take as entanglement measure the Schmidt rank, and determine, for every pair of local dimensions dA and dB, and every r, the largest dimension of a subspace consisting only of entangled states of Schmidt rank r or larger. This exact answer is a significant improvement on the best bounds that can be obtained using the random subspace techniques in Hayden We also determine the converse: the largest dimension of a subspace with an upper bound on the Schmidt rank. Finally, we discuss the question of subspaces containing only states with Schmidt equal to r. © 2008 American Institute of Physics.
Tue, 01 Jan 2008 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1164932008-01-01T00:00:00Z
- On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum stateshttps://scholarbank.nus.edu.sg/handle/10635/116492Title: On the Chernoff distance for asymptotic LOCC discrimination of bipartite quantum states
Authors: Matthews, W.; Winter, A.
Abstract: Motivated by the recent discovery of a quantum Chernoff theorem for asymptotic state discrimination, we investigate the distinguishability of two bipartite mixed states under the constraint of local operations and classical communication (LOCC), in the limit of many copies. While for two pure states a result of Walgate et al. shows that LOCC is just as powerful as global measurements, data hiding states (DiVincenzo et al.) show that locality can impose severe restrictions on the distinguishability of even orthogonal states. Here we determine the optimal error probability and measurement to discriminate many copies of particular data hiding states (extremal d × d Werner states) by a linear programming approach. Surprisingly, the single-copy optimal measurement remains optimal for n copies, in the sense that the best strategy is measuring each copy separately, followed by a simple classical decision rule. We also put a lower bound on the bias with which states can be distinguished by separable operations. © 2008 Springer-Verlag.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1164922009-01-01T00:00:00Z
- Distinguishing Multi-Partite States by Local Measurementshttps://scholarbank.nus.edu.sg/handle/10635/116297Title: Distinguishing Multi-Partite States by Local Measurements
Authors: Lancien, C.; Winter, A.
Abstract: We analyze the distinguishability norm on the states of a multi-partite system, defined by local measurements. Concretely, we show that the norm associated to a tensor product of sufficiently symmetric measurements is essentially equivalent to a multi-partite generalisation of the non-commutative ℓ2-norm (aka Hilbert-Schmidt norm): in comparing the two, the constants of domination depend only on the number of parties but not on the Hilbert spaces dimensions. We discuss implications of this result on the corresponding norms for the class of all measurements implementable by local operations and classical communication (LOCC), and in particular on the leading order optimality of multi-party data hiding schemes. © 2013 Springer-Verlag Berlin Heidelberg.
Tue, 01 Oct 2013 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1162972013-10-01T00:00:00Z
- Distinguishability of quantum states under restricted families of measurements with an application to quantum data hidinghttps://scholarbank.nus.edu.sg/handle/10635/116984Title: Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
Authors: Matthews, W.; Wehner, S.; Winter, A.
Abstract: We consider the problem of ambiguous discrimination of two quantum states when we are only allowed to perform a restricted set of measurements. Let the bias of a POVM be defined as the total variational distance between the outcome distributions for the two states to be distinguished. The performance of a set of measurements can then be defined as the ratio of the bias of this POVM and the largest bias achievable by any measurements. We first provide lower bounds on the performance of various POVMs acting on a single system such as the isotropic POVM, and spherical 2 and 4-designs, and show how these bounds can lead to certainty relations. Furthermore, we prove lower bounds for several interesting POVMs acting on multipartite systems, such as the set of local POVMS, POVMs which can be implemented using local operations and classical communication (LOCC), separable POVMs, and finally POVMs for which every bipartition results in a measurement having positive partial transpose (PPT). In particular, our results show that a scheme of Terhal et. al. for hiding data against local operations and classical communication [31] has the best possible dimensional dependence. © Springer-Verlag 2009.
Sat, 01 Aug 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1169842009-08-01T00:00:00Z
- Improving zero-error classical communication with entanglementhttps://scholarbank.nus.edu.sg/handle/10635/117047Title: Improving zero-error classical communication with entanglement
Authors: Cubitt, T.S.; Leung, D.; Matthews, W.; Winter, A.
Abstract: Given one or more uses of a classical channel, only a certain number of messages can be transmitted with zero probability of error. The study of this number and its asymptotic behavior constitutes the field of classical zero-error information theory. We show that, given a single use of certain classical channels, entangled states of a system shared by the sender and receiver can be used to increase the number of (classical) messages which can be sent without error. In particular, we show how to construct such a channel based on any proof of the Kochen-Specker theorem. We investigate the connection to pseudotelepathy games. The use of generalized nonsignaling correlations to assist in this task is also considered. In this case, an elegant theory results and, remarkably, it is sometimes possible to transmit information with zero error using a channel with no unassisted zero-error capacity. © 2010 The American Physical Society.
Tue, 08 Jun 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1170472010-06-08T00:00:00Z
- Zero-error communication via quantum channels and a quantum Lovász script v sign-functionhttps://scholarbank.nus.edu.sg/handle/10635/117285Title: Zero-error communication via quantum channels and a quantum Lovász script v sign-function
Authors: Duan, R.; Severini, S.; Winter, A.
Abstract: We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain linear space operators as the quantum generalisation of the adjacency matrix, in terms of which the plain, quantum and entanglement-assisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous script v sign function, as the norm-completion (or stabilisation) of a "naive" generalisation of script v sign. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite programme, whose dual we write down explicitly, and that it is multiplicative with respect to the natural (strong) graph product. We explore various other properties of the new quantity, which reduces to Lovász' original script v sign in the classical case, give several applications, and propose to study the linear spaces of operators associated to channels as "non-commutative graphs", using the language of operator systems and Hilbert modules. © 2011 IEEE.
Sat, 01 Jan 2011 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1172852011-01-01T00:00:00Z
- The quantum reverse shannon theorem and resource tradeoffs for simulating quantum channelshttps://scholarbank.nus.edu.sg/handle/10635/117193Title: The quantum reverse shannon theorem and resource tradeoffs for simulating quantum channels
Authors: Bennett, C.H.; Devetak, I.; Harrow, A.W.; Shor, P.W.; Winter, A.
Abstract: Dual to the usual noisy channel coding problem, where a noisy (classical or quantum) channel is used to simulate a noiseless one, reverse Shannon theorems concern the use of noiseless channels to simulate noisy ones, and more generally the use of one noisy channel to simulate another. For channels of nonzero capacity, this simulation is always possible, but for it to be efficient, auxiliary resources of the proper kind and amount are generally required. In the classical case, shared randomness between sender and receiver is a sufficient auxiliary resource, regardless of the nature of the source, but in the quantum case, the requisite auxiliary resources for efficient simulation depend on both the channel being simulated, and the source from which the channel inputs are coming. For tensor power sources (the quantum generalization of classical memoryless sources), entanglement in the form of standard ebits (maximally entangled pairs of qubits) is sufficient, but for general sources, which may be arbitrarily correlated or entangled across channel inputs, additional resources, such as entanglement-embezzling states or backward communication, are generally needed. Combining existing and new results, we establish the amounts of communication and auxiliary resources needed in both the classical and quantum cases, the tradeoffs among them, and the loss of simulation efficiency when auxiliary resources are absent or insufficient. In particular, we find a new single-letter expression for the excess forward communication cost of coherent feedback simulations of quantum channels (i.e., simulations in which the sender retains what would escape into the environment in an ordinary simulation), on nontensor-power sources in the presence of unlimited ebits but no other auxiliary resource. Our results on tensor power sources establish a strong converse to the entanglement-assisted capacity theorem. © 1963-2012 IEEE.
Wed, 01 Jan 2014 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1171932014-01-01T00:00:00Z
- Quantum information: Coping with uncertaintyhttps://scholarbank.nus.edu.sg/handle/10635/117131Title: Quantum information: Coping with uncertainty
Authors: Winter, A.
Wed, 01 Sep 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1171312010-09-01T00:00:00Z
- Nonmalleable encryption of quantum informationhttps://scholarbank.nus.edu.sg/handle/10635/117092Title: Nonmalleable encryption of quantum information
Authors: Ambainis, A.; Bouda, J.; Winter, A.
Abstract: We introduce the notion of nonmalleability of a quantum state encryption scheme (in dimension d): in addition to the requirement that an adversary cannot learn information about the state, here we demand that no controlled modification of the encrypted state can be effected. We show that such a scheme is equivalent to a unitary 2-design [Dankert, e-print arXiv:quant-ph/0606161], as opposed to normal encryption which is a unitary 1-design. Our other main results include a new proof of the lower bound of (d2 -1) 2 +1 on the number of unitaries in a 2-design [Gross, J. Math. Phys. 48, 052104 (2007)], which lends itself to a generalization to approximate 2-design. Furthermore, while in prime power dimension there is a unitary 2-design with d5 elements, we show that there are always approximate 2-designs with O (-2 d4 log d) elements. © 2009 American Institute of Physics.
Thu, 01 Jan 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1170922009-01-01T00:00:00Z
- On the speed of fluctuations around thermodynamic equilibriumhttps://scholarbank.nus.edu.sg/handle/10635/117105Title: On the speed of fluctuations around thermodynamic equilibrium
Authors: Linden, N.; Popescu, S.; Short, A.J.; Winter, A.
Abstract: We study the speed of fluctuations of a quantum system around its equilibrium state, and show that the speed is extremely small at almost all times in typical thermodynamic cases. This suggests an alternative view on the nature of thermal equilibrium and, in particular, of the origin of thermal fluctuations. We argue that instead of equilibrium being a dynamical process in which the system actively fluctuates in time, the fluctuations are due to quantum uncertainties in an essentially static state. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.
Fri, 28 May 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1171052010-05-28T00:00:00Z
- Quantum mechanical evolution towards thermal equilibriumhttps://scholarbank.nus.edu.sg/handle/10635/117132Title: Quantum mechanical evolution towards thermal equilibrium
Authors: Linden, N.; Popescu, S.; Short, A.J.; Winter, A.
Abstract: The circumstances under which a system reaches thermal equilibrium, and how to derive this from basic dynamical laws, has been a major question from the very beginning of thermodynamics and statistical mechanics. Despite considerable progress, it remains an open problem. Motivated by this issue, we address the more general question of equilibration. We prove, with virtually full generality, that reaching equilibrium is a universal property of quantum systems: almost any subsystem in interaction with a large enough bath will reach an equilibrium state and remain close to it for almost all times. We also prove several general results about other aspects of thermalization besides equilibration, for example, that the equilibrium state does not depend on the detailed microstate of the bath. © 2009 The American Physical Society.
Thu, 04 Jun 2009 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1171322009-06-04T00:00:00Z
- Tensor rank and stochastic entanglement catalysis for multipartite pure stateshttps://scholarbank.nus.edu.sg/handle/10635/117175Title: Tensor rank and stochastic entanglement catalysis for multipartite pure states
Authors: Chen, L.; Chitambar, E.; Duan, R.; Ji, Z.; Winter, A.
Abstract: The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state |W3=1√3(|100+|010+|001) and its N-partite generalization |WN. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of |W3 have a rank of either 15 or 16, (ii) two copies of |WN have a rank of 3N-2, and (iii) n copies of |WN have a rank of O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple-copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states. © 2010 The American Physical Society.
Mon, 08 Nov 2010 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1171752010-11-08T00:00:00Z